1. Fundamental Concepts 1.1 Introduction Theoretical framework of classical physics: ∗ state (at fixed t) is defined by a point {xi , pi } in phase space � �∂F ∂G ∂F ∂G � (flat space, symplectic ∗ Poisson bracket {xi , pj } = δij {F G} = i ∂x i ∂pi − ∂pi ∂xi mfld) ∗ Observables are functions O(xi , pj ) on phase space [ex: x2 + y 2 + z 2 , p2 /2m, . . .] ∗ Hamiltonian H(x, p) defines dynamics q̇ = {q, H} for q = xi , pj , . . . � � � � � � I B I � (x, p) qM � � � z� � phase space qM Framework describes all of mechanics. E & M including fluids, materials, etc. . . . • Simple, intuitive conceptual framework • Deterministic • Time-reversible dynamics Example: Classical simple harmonic oscillator (SHO) H= 1 2 k 2 p x + 2 2m p � ! ' � $ 2 X � X " x 1 p m ṗ = {p, H} = −kx (= mẍ) ẋ = {x, H} = Theoretical framework of quantum physics ∗ state defined by (complex) vector |v� in vector space H. (Hilbert space) ∗ Observables are Hermitian operators Odagger = O ∗ Dynamics: |v(t)� = e−iHt/�|v(0)� (H hermitian, � constant) ∗ “Collapse postulate” [(subtleties: (simple version) � If |φ� = αi |λi�, � λj with A|λi � = λi |λi �, λi = 2 Then with prob. |αi | , measure A = λi , |φ� = | λi � after measurement degeneracy, cts. spectrum)] This framework [(with suitable generalizations, i.e. field theory)] describes all quantum sys­ tems, and all experiments not involving gravitational forces. – Atomic spectra (quantization of energy) – Semiconductors transistors, (quantum tunneling) • Counterintuitive conceptual framework • Nondeterministic • Irreversible dynamics Both classical & quantum conceptual frameworks useful in certain regimes. Classical picture not fundamental — replaced by QM. Is QM fundamental? Perhaps not, but describes all physics of current relevance to technology & society. Simple example of QM: 2-state system (spin 1 2 3 particle) Stern & Gerlach 1922 S � � � --- � � � �' � � � � � � � � � � -Ag atoms Oven �� I�B � � N screen Silver: 47 electrons, angular momentum =⇒ µ (mag. dipole moment) from spin of 47th electron. U = −µ · B ∂U ∂Bz Fz = − = µz ∂Z ∂Z Gives force along ẑ depending on µz . Energy Classically, expect & Actually see # % # % 4 e e� ≈ Sz 2mc mc � [� = 1.0546 × 10−27 ergs] = ± 2 µz ≈ ± Sz So measuring Sz =⇒ discrete values (2 states) Can build sequential S − G experiments, using components Sz = + splitter Z Sz = − ⎝ � � � � � � � � � � � � � filters � � � � � � � � � � � � ⎜ Z Sz = + � � � � � � � � � � � � � � � � Z Sz = − (Can also form splitter, filters on x-axis, etc.) Single particle experiments i) Sz =+ Z ii) � � � � � � � � Sz= + Z Sz =+ Z � � � � � � � � Sx= + X Sx = − 5 100% 0% 50% 50% iii) Sz =+ Z Sx= + X � � � � � � � � Sz= + Z Sx = − Sz = − 100% 0% BUT iv) Sz =+ Z v) Sx= + X � � � � � � � � � � � � � � � � Sz =+ Z � � � � � � � � � � � � � � � � X Sx = − Sz= + Z Sz = − Sz= + Z Sz = − 50% 50% 50% 50% Cannot simultaneously measure Sz , Sx “incompatible observables” S z Sx = � Sx S z Analogous to 2-slit experiment for photons. In iii), not measuring Sx . in iv), v) measuring Sx . iii) needs “interference” of probability wave — no classical interpretation. iii), iv) =⇒ Irreversible, nondeterministic dynamics (assuming locality) 1.2 1.2.1 Mathematical Preliminaries Hilbert spaces First postulate of QM: ∗ The state of a QM system at time t is given by a vector (ray) |α� in a complex Hilbert space H. [[will state more precisely soon.]] Vector spaces A vector space V is a collection of objects (“vectors”) |α� having the following proper­ ties: 6 A1: |α� + |β� gives a unique vector |γ� in V . A2: (commutativity) |α� + |β� = |β� + |α� A3: (associativity) (|α� + |β�) + |δ� = |α� + (|β� + |δ�) A4: ∃ vector |φ� such that |φ� + |α� = |α� ∀|α� A5: For all |α� in V , −|α� is also in V so that |α� + (−|α�) = |φ�. [(A1–A5): V is a commutative group under +] For some Field F (i.e., R, C, with +, ∗ defined) scalar multiplication of any c ∈ F with any |α� ∈ V gives a vector c|α� ∈ V . Scalar multiplication has the following properties M1: c(d|α�) = (cd)|α� M2: 1|α� = |α� M3: c(|α� + |β�) = c|α� + c|β� M4: (c + d)|α� = c|α� + d|α�. V is called a “vector space over F .” F = R: “real v.s.” F = C: “complex v.s.” Examples of vector spaces a) Euclidean D-dimensional space is a real v.s. � a1 � .. . a D b) State space of spin 12 particle is 2D cpx v.s. states: c+ |+� + c− |−�, c± ∈ C [[note: certain state are physically equivalent]] c) space of functions f � : [0, L] → C Henceforth we always take F = C Subspaces V ⊂ W is a subspace of a v.s. W if V satifies all props of a v.s. and is a subset of W . [suffices for V to be closed under +, scalar mult.] Ray A ray in V is a 1D subspace {c|α�}. 7 Linear independence & bases |α1 �, |α2 �, . . . , |αn � are linearly independent iff c1 |α1 � + c2 |α2 � + · · · + cn |αn � = 0 has only c1 = c2 = · · · = cn = 0 as a solution. If |α1 �, . . . , |αn � in V are linearly independent, but all sets of n + 1 vectors are linearly dependent, then • V is n-dimensional (n may be finite, countably ∞, or uncountably ∞.) • |α1 �, . . . , |αn � form a basis for the space V . If |α1 �, . . . , |αn � form a basis for V , then any vector |β� can be expanded in the basis as � ci |αi � . (Thm.) |β� = i Unitary spaces A complex vector space V is a unitary space (a.k.a. inner product space) if given |α�, |β� ∈ V there is an inner product �α|β� ∈ C with the following properties: I1: �α|β� = �β|α� ∗ I2: �α|(|β� + |β ��) = �α|β� + �α|β �� I3: �α|(c|β�) = c�α|β� I4: �α|α� ≥ 0 I5: �α|α� = 0 iff |α� = |0� �α|β� is a sesquilinear form (linear in β), conj. lin. in |α� Examples: � z1 � .. . , zi ∈ C. zN ∗ wN �z|w� = zI∗ w1 + z2∗ w2 + · · · + zN a) V = CN , N-tuples |z� = b) V = {f : [0, L] → C} �L �f |g� = 0 f ∗ (x)g(x) dx Terminology: ⎧ �α|β� = norm of |α� (sometimes, �|α�� if �α|β� = 0, |α�, |β� are orthogonal 8 Dual Spaces For a (complex) vector space V , the dual space V ∗ is the set of linear functions �β| : V → C. Given the inner product �β|α�, can construct isomorphism. |α� ←→ �α| V ←→ V ∗ note: c|α� ←→ c∗ �α| . Physics notation (Dirac ”bra-ket” notation) |α� ∈ V �β| ∈ V ∗ ket bra Hilbert space A space V is complete if every Cauchy sequence {|αn �} converges in V ∀�∃N : �|αn � − |αm �� < � ∀m, n > N (i.e., ∃|α� : limn→∞ �|α� − |αn �� = 0.) A complete unitary space is a (complex) Hilbert space. Note: an example of an incomplete unitary space is the space of vectors with a finite number of nonzero entries. The sequence { (1, 0, 0, . . . ), (1, 12 , 0, 0, . . . ), (1, 12 , 13 , 0, . . . ) ... } is Cauchy, but doesn’t converge in V . This is not a Hilbert space. A Hilbert space can be: a) Finite dimensional (basis |α1 �, . . . , |αn �) Ex. spin 12 particle in Stern-Gerlach expt. b) Countably infinite dimensional (basis |α1 �, |α2 �, . . .) Ex. Quantum SHO c) Uncountably infinite dimensional (basis |αx �, x ∈ R) [Technical aside: ¯ =V. A space V is separable if ∃ countable set D ⊂ V so that D (D dense in V : ∀�, |x� ∈ V ∃|y� ∈ D : �|x� − |y�� < � ) (a), (b) are separable, (c) is not. non-separable Hilbert spaces are very dicey mathematically. Generally, separability implicit in discussion — e.g., label basis |αi �, i takes discrete values ] 9 Orthonormal basis An orthonormal basis is a basis |ϕi � with �ϕi|ϕj � = δij Any basis |α1 �, |α2 �, . . . can be made orthonormal by Schmidt orthonormalization |α1 � |φ1 � = ⎧ �α1 |α1 � � |α2 � = |α2 � − |φ1 ��φ1 |α2 � |α� � |φ2 � = ⎧ �2 � �α2 |α2 � � |α3 � = |α3 � − |φ1 ��φ1 |α3 � − |φ2��φ2 |α3 � .. . Gives |φi� with �φi|φj � = δij . If {|φi�} are an orthonormal basis then for all |α�, � ci |φi � , ci = �φi |α� . |α� = i Can write as |α� = � i |φi ��φi |α�. (Completeness relation) Schwartz inequality �α|α��β|β� ≥ |�α|β�|2 ∀|α�, |β� . Proof: write |γ� = |α� + λ|β� → �γ|γ� = �α|α� + λ�α|β� + λx �β|α� + |λ|2�β|β� −�β|α� set λ = �β|β� |�β |α�| �γ|γ� = �α|α� − ≥0 �β|β� � [Second postulate of QM: ∗ Observables are (Hermitian) operators on H [self-adjoint] 10 1.2.2 Operators Linear operators A linear operator from a VS V to a VS W is a transformation such that A|α� + |β� = A|α� + A|β� ∀|α�, |β� . We write A = B iff A|α� = B|α� ∀|α�. A acts on V ∗ through (�β| ∀A)|α� = �β|(A|α�) (acts on right on bras.) Outer product A simple class of operators are outer products |β��α| (|β��α|)|γ� = |β��α|γ� Adjoint Recall correspondence V ↔ V∗ |α� ↔ �α| Given an operator A, define A† (adjoint of A) (Hermitian conjugate) by A† |α� ↔ �α|A Example (|β��α|)† = �α|�β|. Follows that �α|A†|β� = (�β|A|α�)∗. Hermitian operators A is Hermitian if A = A† (∼ self-adjoint) ⎛ Technical aside: mathematically, Hermitian called ”symmetric”. Self-adjoint iff symmetric +A & A† have same domain of definition, relevant to i.e., Dirac op. in monopole background (symmetric op. with several self-adjoint extensions) more: Reed & Simon, Jackiw �0 2 Example of domain of def: Consider H = L2 (R) = {funs f : R → C : −∞ f ∗ f < 0 e−x ∈ H, 2 O = mult by ex ⎞ 2 2 Oe−x ∈ / H, so e−x not in domain D(O). Linear operators A form a vector space under addition (+ commutative, associative) (A + B)|α� = A|α� + B|α� Mult. defined by (AB)|α� = A(B|α�) Generally AB = � BA But (AB)C = A(BC) Note: (XY )+ = Y + X + 11 Identity operator: 11 11|α� = �α|. � Functions of one operator f (A) = Cn An can be expanded as power series (must be careful outside ROC — can do in general if diagonalizable) Diagonalizable operators: can always compute f (A) if f defined for diagonal elements (eval­ ues) Functions f (A, B) must have definable ordering prescription. (e.g. eA Be−A = B + AB − BA + · · · ) Inverse A−1 satisfies AA−1 = A−1 A = 11 Does not always exist. (Ex. if A has an ev. = 0.) Note: BA = 11 does not imply AB = 11 (Ex. later) Isometries U is an isometry if U + U = 11, since preserves inner product (�β|U + )(U|α�) = �β|α� Unitary operators U is unitary if U † = U −1 . Example: non-unitary isometries. (Hilbert Hotel) Consider the shift operator S|n� = |n + 1� acting on H with countable on basis {|n�, = 0, 1, . . . } S = |n + 1��n| satisfies S + S = 11 but not SS + = 11. (SS + = 11 − |0��0|). S has no (2-sided) inverse. Projection operators A is a projection if A2 = A. Ex. A = |α��α| for �α|α� = 1. Eigenstates & Eigenvalues If A|α� = a|α� then |α� is an eigenstate (eigenket) of A and a is the associated eigenvalue. Spectrum The spectrum of an operator A is its set of eigenvalues {a} [technical aside: this is the “point spectrum”, mathematically, spectrum of A = set of λ : A − λ11 is not invertible] 12 Important theorem If A = A† , then all eigenvalues ai of A are real, and all eigenstates associated with distinct ai are orthogonal. Proof: A|a� = a|a� ⇒ �a|A† = �a|a∗ ⇒ �b|(A − A+ )|a� = (a − b∗ )�b|a� = 0 if if a = a∗ is real. �b|a� = 0 a = b, a = � b , � Consequence of theorem: For any Hermitian A, can find an O.N. set of eigenvectors |ai � (ai not necc. distinct — can be degenerate) A|ai � = ai |ai � , [Proof: use Schmidt orthog. for each subspace of fixed evalue a — OK as long as countable # of (indep.) states for any a (e.g. in separable H) [caution: this set spans space of eigenvectors, but may not be complete basis] Completeness relation If φi are a complete on basis for H. |α� = � |φi ��φi|α� ∀|α� , i � (completeness) so i |φi ��φi | = 11 (sum of projections onto 1D subspaces) Matrix and vector representations If H is separable, ∃ a countable on � basis,⎪|φi � �φ1 |α� � ��φ2 |α�� can write |α� = i |φi��φi |α� ⇒ � � .. . �β| = � �β|φi��φi| ⇒ (�β|φ1��β|φ2� · · · ) i 13 ⎪ � �φ |A|φ � �φ |A|φ � · · · 1 1 1 2 � � � A= |φi ��φi|A|φj ��φj | ⇒ ��φ2 |A|φ1� �φ2 |A|φ2 � · · ·� .. .. .. i,j . . . If �ai | are a basis of O.N. eigenvectors w.r.t. A, �ai |A|aj � = ai δij � ⎪ a1 � � � a2 � � � A= |ai �ai �ai | ⇒ � � � a3 � � .. . Usual matrix interpretation of adjoint, dual correspondence (adjoint = conjugate transpose) �φi|A|φj � = �φj |A† |φi �∗ � ⎪ c1 �c2 � dual dual: |α� ⇒ � � −→ �α| ⇒ (c∗1 c∗2 · · · ) .. . � �α|β� ⇒ c∗i di inner product. � ⎪ d � ∗ ∗ � � 1� c1 c2 · · · �d2 � .. . When do eigenvectors of A = A† form a complete basis for H? True when H is finite dimensional (explicit construction from diagonalization), not neces­ sarily when H infinite dimensional. Defs. A is bounded iff sup |α�∈H |α��=|0� �α|A|α� �α|α� < ∞ A is compact if every bounded sequence {|αn �} (�αn |αn � < β) has a subsequence {|αnk �} so that {A|αnk �} is norm convergent in H. Facts: • A compact ⇒ A bounded. • Every compact A = A† has a complete set of eigenvectors. (compactness sufficient) • not necessarily true for bounded A = A† . (neither necessary nor sufficient) Ex. H = L2 ([0, 1]) A = x. √ A is bounded, not compact. (|αn � = xn 2n + 1) A has no eigenvectors in H. For physics: Only interested in operators with a complete set of eigenvectors. These are called observables. Observables need not be bounded or compact. (note: will reverse this stance a bit for cts systems!) 14 Trace The trace of an operator A is Tr A = � �φi|A|φi � , i = � ai = i � |φi� ON basis Aii . i (ai = eigenvalues of A) Unitary transformations If |ai �, |bi � are two complete ON bases, [(Ex. eigenkets of 2 Hermitian operators)] can define U so that U|ai � = |bi � (since |ai � a basis defines U on all of H). so �bi | = �ai |U † . � U = U11 = U i |ai ��ai | � = i |bi ��ai | � U† = i |ai ��bi | � � so UU † = i,j|bi ��ai |aj ��bj | = δij ij |bi ��bj | = 11 and UU † = 11, so U −1 = U + , U unitary. We have – Analogous to rotations in Euclidean 3-space M : M + M = MM + = 11. U are symme­ tries of H. Unitary transforms of vectors & operators A vector |α� has representations in two bases as � � di |bi � . |α� = ci |ai � = How are these related? � dj |bj � = � dj U|aj � j = � dj |ai ��ai |U|aj � i,j so ci = Uij dj , Uij = �ai |U|aj � are mtx elements of U in a rep. Similarly, X = |ai �Xij �aj | = |bk �Yk� �b� | gives Xij = Uik Yk� Ulj† . 15 Diagonalization of Hermitian operators Theorem. A Hermitian matrix (finite dim) Hij = �φi |H|φj � can always be diagonalized by a unitary transformation. Proof. if |φi� a general ON basis, ON eigenvectors |hi � related to |φi � through unitary . |hi � = U |φi � , �hi |H|hj � = δij hi = �φi|U † HU |φj � + Hk� U�j is diagonal. so Uik (generalizes to any observable) Algorithm for explicit diagonalization of a matrix H (finite dimensional): 1) Solve det(H − λ11) = 0 for N × N matrices, N solutions are eigenvalues of H. 2) Solve Hij cj = λci for ci ’s for each λ. N linear eqns. in N unknowns. Gives eigenvalues & eigenvectors. Invariants Some functions of an operator A are invariant under U : � Tr A = �φi |A|φi� , |φi � ON basis � † Tr U † AU = Uij Ajk Ukj = δjk Ajk = Tr A i,j,k [Technical note: careful for ∞ matrices — need all sums converging.] Another invariant: det A: det(U † AU ) = det U det A det U † = det UU † det A = det A [Note: full spectrum of ev’s is invariant!] Simultaneous diagonalization Theorem. Two diagonalizable operators A, B are simultaneously diagonalizable iff [A, B] = 0 ⇒ say A|αi � = ai |αi � , B|αi � = bi |αi � AB|αi � = BA|αi � = ai bi |αi � . ⇐ Say AB = BA , A|αi � = ai |αi � . AB|αi � = ai B|αi � , so B keeps state in subspace of e.v. ai . Thus, B is block-diagonal, can be diagonalized in each ai subspace � 16 1.3 The rules of quantum mechanics [[Developed over many years in early part of C20. Cannot be derived — justified by logical consistency & agreement with experiment.]] 4 basic postulates: 1) A quantum system can be put into correspondence with a Hilbert space H so that a definite quantum state (at a fixed time t) corresponds to a definite ray in H. so |α� ≈ c|α� represent same physical state convenient to choose �α|α� = 1, leaving phase freedom eiφ |α� • Note: still a classical picture of state space (“realist approach”). Path integral approach avoids this picture. • “state” really should apply to an ensemble of identically prepared experiments (“pure ensemble” = pure state.) Ex. states coming out of SG filter Sz = + Z |α� = |+� . � � � � � � � � 2) Observable quantities correspond to Hermitian operators whose eigenstates form a complete set. Observable quantity = something you can measure in an experiment. [[Note: book constructs H from eigenstates of A: logic less clear as HA �= HB for some A, B.]] 3) An observable H = H † defines the time evolution of the state in H through i� d |ψ(t + Δt)� − |ψ(t)� |ψ(t)� = i� lim = H|ψ(t)� . Δt→0 dt Δt (Schrödinger equation) 17 4) (Measurement & collapse postulate) If an observable A is measured when the system is in a normalized state |α�, where A has an ON basis of eigenvectors |ai � with eigenvalues ai . a) The probability of observing A = a is � |�aj |α�|2 = �α|Pa |α� j:aj =a where Pa = � j:aj =a |aj ��aj | is the projector onto the A = a eigenspace. b) If ⎧ becomes |αa � = Pa |α� = �A = a is observed, after the measurement the state |a ��a |α� (normalized state is |α ˜ � = |α �/ �αa |αa �). j a a j:aj =a j Discussion of rule (4): Simplest case: nondegenerate eigenvalues � |α� = ci |ai � , ai = � aj . Then probability of getting A = ai is |ci |2 . � 2 Norm of �α|α� = 1 ⇔ |ci | = 1. After measuring A = ai , state becomes |α ˜ i � = |ai �. This postulate involves an irreversible, nondeterministic, and discontinuous change in the state of the system. – source of considerable confusion – less troublesome picture: path integrals. – alternatives: non-local hidden variables (’t Hooft?), string theory — new principles (?) For purposes of this course, take (4) as fundamental, though counterintuitive, postulate. To discuss probabilities, need ensembles. Consequence of (4): Expectation value of an observable A in state |α� is � � 2 |ai �ai �ai |. |ci | ai = �α|A|α� since A = �A� = i i So for: 4 basic postulates of Quantum Mechanics: 18 1) State = ray in H [incl. def of H space] 2) Observable = Hermitian operator with complete set of eigenvectors 3) i� dtd |ψ(t)� = H|ψ(t)� 4) Measurement & collapse Probability A = a: �α|P a|α� ⎧ After measurement, system −→ |α ˜ a � = Pa |α�/ �α|Pa |α� � Pa = |aj ��aj | j:aj =a � ⇒ Expectation value of A: �A� = �α|A|α� = � |ci |2 ai if |α� = � ci |ai � � 4⇒ These are the rules of the game. Rest of the course: Examples of physical systems, tools to solve problems. An example revisited in detail Back to spin- 12 system. ⎝ State space � � � � H = {|α� = C+ |+� + C− |−� , C± ∈ C} � P 1 Unit norm condition � � �α|α� = |C+ |2 + |C− |2 = 1 � � ⎜ eiθ |α� , |α� are physically equivalent Operators: Sz = � σ 2 z = � 2 Sx = � σ 2 x = � 2 Sy = � σ 2 y = � 2 � � 1 0 0 −1 0 1 � measures spin along z-axis � measures spin along x-axis �1 0 � 0 −i i 0 measures spin along y-axis For general axis n̂: Sn = S · n̂ (HW #2) ± �2 . has eigenvalues eigenstates |Sn ; ±� : Sn |Sn ; ±� = ± �2 |Sn ; ±� form complete basis. 19 |Sx ; ±� = √12 |+� ± |Sy ; ±� = √12 |+� ± |Sz ; ±� = |+� √1 |−� 2 √i |−� 2 ⎠ � ⎟ in Sz basis. Some further properties of Si : [Si , Sj ] = i�ijk �Sk {Si , Sj } = Si Sj + Sj Si = 12 �2 δij 3 3� 2 � 1 0 � S 2 = S · S = Sx2 + Sy2 + Sz2 = �2 11 = 0 1 4 4 And [S 2 , Si ] = 0 . Measurement If |α� = c+ |+� + c− |−�, �α|α� = 1, prob. that S z = prob. that Sz = +� 2 −� 2 is | c+ |2 is | c− |2 Consider single particle experiments from lecture 1. Sz = + �2 i) Sz = + �2 Sz � � � � � � � � Sz 100% 0% Sz = − �2 |α� = |+� Repeated measurement of Sz gives + �2 100% of the time. Sz = + ii) Sz � � � � � � � � Sx |α� = |+� |Sx ; ±� = so so √1 (|+� ± |−�) 2 |α� = |+� = √1 2 [|Sx ; +� + |Sx ; −�] prob. Sx = + �2 is 12 (50%) prob. Sx = − �2 is 12 (50%) 20 Sx = + Sx = − 50% 50% iii) S z = + Sx = + Sz Sz = + � � � � � � � � � � S x |α� = |+� Sz � �B S x = − � � � 1 100% 0% S z = − |α− � = 2 (|+� − |−�) � � |α+ � = 12 (|+� + |−�) |α� = |α+ � + |α− � = |+� Combined state |α� = |+� enters last measurement apparatus, since Sx not measured. Gives Sz = + �2 100% of time. iv) Sz = + Sx = + Sz � � � � � � � � √1 2 �� � � � � � � � � � � S x |α+ � = |α� = |+� state |α+ � = Sz = + √1 2 Sz � (|+� + |−�) 50% 50% Sz = − (|+� + |−�) enters last apparatus. Prob. Sx = + �2 : (50%) Prob. Sx = − �2 : (50%) Compatible vs. incompatible observables Observables A, B are: Compatible if [A, B] = AB − BA = 0 incompatible if [A, B] = � 0. Examples: S 2 , Sz Sx , Sy are compatible are not compatible. Theorem. Compatible observables A, B can be simultaneously diagonalized, and have eigen­ vectors |ai , bi � with A|ai , bi � = ai |ai , bi � B|ai , bi � = bi |ai , bi � . 21 (Proof in last lecture: AB|α� = aB|α� if A|α� = a|α� so B = Ha → Ha , diagonalize in each block.) A complete set of commuting observables (CSCO) is a set of observables {A, B, C, . . .} such that all observables in the set commute: [A, B] = [A, C] = [B, C] = · · · = 0 and such that for any a, b, . . . at most one solution exists to the eigenvalue equations A|α� = a|α� B|α� = b|α� .. . Tensor product spaces � useful for many-particle systems [+ quantum computing, . . . ]bigr) Given two Hilbert spaces H(1) , H(2) , with complete ON bases |φ(1) �i , |φ(2) �j , the tensor product H = H(1) ⊗ H(2) is the Hilbert space with ON basis (1) (2) |φij � = |φi � ⊗ |φj � and inner product (1) (1) (2) (2) �φi,j |φk,�� = �φi |φk �1 , �φj |φ� �2 . If H(1) , H(2) have dimensions N, M, then H = H(1) ⊗ H(2) has dimension NM. If H(1) , H(2) separable, H is separable. [in particular, if either or both of H(1) , H(2) have countable basis & both countable or finite H has countable basis.] Tensor product of kets and operators Kets: If |α� = |β� = � � (1) ci |φi � ∈ H(1) (2) di |φj � ∈ H(2) 22 are kets in H(1) , H(2) . then |α� ⊗ |β� = � ci dj |φi,j � ∈ H i,j is in H = H(1) ⊗ H(2) . [Note: not all vectors in H are of tensor product form. Ex. |φ1,1 � + |φ1,2 � + |φ2,1�] Operators: If A, B are operators on H(1) , H(2) , then we can construct A ⊗ B as an operator on H = H(1) ⊗ H(2) through (1) (2) (A ⊗ B)|φi,j � = (A|φi �) ⊗ (B|φj �) . [defines A ⊗ B on all of H by linearity] If A, B are observables, then A ⊗ B is an observable. Summary of Tensor product spaces H = H(1) ⊗ H(2) dim H = (dim H(1) )(dim H(2) ) (2) Basis: |φi,j � = |φ(1) � ⊗ |φj � (2) �φi,j � = �φ(1) | ⊗ �φj | � � (1) (2) dj |φj � Kets: |α� Ci |φi �|β� = � |α� ⊗ |β� = Ci dj |φi,j � Bras same � � (1) (2) Ci,j (A|φi �) ⊗ (B|φj �) Operators (A ⊗ B) Ci,j |φi,j � = Simple class of operators on H: A ⊗ 11 , 11 ⊗ B . If A, B act on H(1) , H(2) , will often refer to these as just A, B when context is clear. Useful relation: (A ⊗ B) · (C ⊗ D) = (AC) ⊗ (BD) . 23 Note: [(A ⊗ 11), (11 ⊗ B)] = 0. Notation: in many books, tensor product symbol is omitted |α� ⊗ |β� ⇒ |α�|β� A ⊗ B ⇒ AB . CSCO’s in tensor product spaces If {A1 , A2 , . . . , Ak } are a CSCO for H(1) , & {B1 , . . . , B� } are a CSCO for H(2) , then {A1 , . . . , Ak , B1 , . . . B� } are a CSCO for H(1) ⊗ H(2) [[Ex. of notation A1 = A1 ⊗ 11.]] Example of tensor products: Two spin- 12 particles (1) (2) Consider two spin- 12 particles with Hilbert spaces H2 , H2 . The two-particle Hilbert space is (1) (2) H = H2 ⊗ H2 . A basis for H is: |++� |+−� |−+� |−−� = = = = |+�1 ⊗ |+�2 |+�1 ⊗ |−�2 |−�1 ⊗ |+�2 |−�1 ⊗ |−�2 Operators: A complete set of commuting observables is Sz(1) = Sz(1) ⊗ 11 Sz(1) = 11 ⊗ Sz(2) . 24 Consider operators Sz = Sz(1) + Sz(2) . ⎪ � � 0 0 0 �0 0 0 0 � � = � �0 0 0 0 � 0 0 0 −� Sz(1) Sz(2) = (S (1) )z ⊗ 11)(11 ⊗ S (2) ) = Sz(1) ⊗ Sz(2) �� � = � � 2 � 2 ⎪ �� − �2 �� �� �� ⎪ 2 − �2 − �2 ⎪ � +1 � �2 � −1 = � −1 4 � 2 � � � − �2 � �. � +1 Incompatible observables If [A, B] = � 0, then cannot simultaneously diagonize A, B . Experiments A= a A � � � � � � � � � � � � � � B B= b1 B= b2 ... B = bk (Assume A, B, C nondegenerate) i) Allow all bi to combine without measuring B Probability (C = c) = |�c|a�|2 [B not measured] 25 C= c C ii) Measure Bi & allow all parts to combine Probability (B = bi ) = |�bi |a�|2 Prob. (C = c given B = bi ) = |�c|bi �|2 � Prob. (C = c) = |�c|bi �|2 |�bi |a�|2 i [when B measured] = � �c|bi ��bi |a��a|bi ��bi |c� i = � zi∗ zi , zi = �a|bi ��bi |c� i know ( � � zi )( zi∗ ) = |�a|c�|2 . So prob. (C = c) does not depend on measurement of B when � � � zi∗ zi = ( zi∗ )( zj ) . i i j Sufficient condition: only one zi � = 0, so either �a|bi � = 0 or �c|bi � = 0 for all but one value of i Sufficient condition: either [A, B] = 0 or [B, C] = 0. Dispersion For A an observable, |α� a state, define ΔA = A − �A� �ΔA2 � is dispersion of A. �ΔA2 � = �A2 − 2A�A� + �A�2 � = �A2 � − �A�2 is variance (a.k.a. mean square deviation) of A. If A|α� = a|α� �ΔA2 � = a2 − a2 = 0 Variance measures “fuzziness” of state. 26 Example: In state |+� � � �� 1 = �2 0 4 � � ��2 � � � � � 1 = − 1 0 2 0 − �2 � �ΔSz2 � � � 1 0 � �2 4 �2 �2 − = 0. 4 4 �� � � � � �2 1 2 4 �ΔSz � = 1 0 �2 0 4 � � ��2 � � � � 0 � 1 2 = − 1 0 � 0 0 2 = �2 . = 4 Uncertainty relation If A, B are observables, �ΔA2 ��ΔB�2 ≥ 14 |[A, B]|2 Proof. Schwartz: (on ΔA|α�,ΔB|α�) (�α|ΔA)(ΔA|α�)(�α|ΔB�)(ΔB|α�) ≥ (�α|ΔA)(ΔB|α�)(�α|ΔB)(ΔA|α�) � � � �2 = ��α| 12 [ΔA, ΔB] + 12 {ΔA, ΔB} |α�� � �[A, B]� + �{ΔA, ΔB}� = 14 � ↑ imaginary ([A, B] skew-Hermitian) [prob. 1–1.] real ({ΔA,ΔB} Hermitian) = 1 4 |�[A, B]�|2 + 41 |�{ΔA, ΔB}�|2 ≥ 1 4 |�[A, B]�|2 . � Example: In state |α� = |+�. �ΔSz2 � = 0 �2 �ΔSx2 � = 4 27 ↑ �2 � 1 4 1.4 |�[Sz , Sx ]�|2 = 1 4 |�Sy �|2 = 0 . Position, momentum and translation Until now, all explicit examples involved finite-dimensional matrices. Generalize to continuous degrees of freedom. Want to describe particle in 3D by wavefunction ψ(x, y, z) Simply to 1D: ψ(x) Want |ψ(x)|2 dx = probability particle is in region dx. � � � � � � � � � �� �� �� ��� ⎨⎩⎫⎬ dx |ψ(x)|2 x Natural Hilbert space: L2 (R): �∞ Square integrable functions −∞ |ψ(x)|2 < ∞. [[To precisely define, need Lebesgue measure, . . . ]] Can do QM in this framework. L2 (R) is a separable Hilbert space. Typical observables on L(2) (R): P[a,b] projection on interval [a, b] � f (x) , a≤x≤b (P[a,b] f )(x) = 0, otherwise Quote from Von Neumann: Dirac, in several papers, as well as in his recently published book, has given a representation of quantum mechanics which is scarcely to be surpassed in brevity and elegance, and which is at the same time of invariant character. It is therefore perhaps fitting to advance a few arguments on behalf of our method, which deviates considerably from that of Dirac. The method of Dirac, mentioned above, (and this is overlooked today in a great part of quantum mechanical literature, because of the clarity and elegance of the theory) 28 in no way satisfies the requirements of mathematical rigor — not even if these are reduced in a natural and proper fashion to the extent common elsewhere in theoretical physics. For example, the method adheres to the fiction that each self-adjoint operator can be put in diagonal form. In the case of those operators for which this is not actually the case, this requires the introduction of “improper” functions with selfcontradictory properties. The insertion of such a mathematical “fiction is frequently necessary in Dirac’s approach, even though the problem at hand is merely one of calculating numerically the result of a clearly defined experiment. There would be no objection here if these concepts, which cannot be incorporated into the present day framework of analysis, were intrinsically necessary for the physical theory. Thus, as Newtonian mechanics first brought about the development of the infinitesimal calculus, which, in its original form, was undoubtedly not self-consistent, so quantum mechanics might suggest a new structure for our “analysis of infinitely many variables” — i.e., the mathematical technique would have to be changed, and not the physical theory. But this is by no means the case. It should rather be pointed out that the quantum mechanical “Transformation theory” can be established in a manner which is just as clear and unified, but which is also without mathematical objections. It should be emphasized that the correct structure need not consist in a mathematical refinement and explanation of the Dirac method, but rather that it requires a procedure differing from the very beginning, namely, the reliance on the Hilbert theory of operators. 29